When the distances of the unit circle are worked out for a set of Euler angles, what is this result called? Is this a 'positional' vector? A generic vector?

I would just like it spelled out how we go from Euler angles to distances. I know movement can be worked out with trig fx, but I want to (and have been advised to use) matrices for this.

What "distances"? Are you referring to the components of a vector in Cartesian coordinates?

A set of Euler angles corresponds to a rotation (orthonormal matrix) rather than a direction (vector). A rotation requires three parameters to specify (e.g. yaw, pitch, roll), a direction only requires two (e.g. heading and elevation, which roughly correspond to yaw and pitch).

To convert Euler angles to a matrix, you need to be clear on which axis is which (is the Z axis up or forward? Or backward?), the order in which the rotations are applied, and the sign of each rotation (is a positive angle clockwise or anticlockwise?). There is no universal "correct" answer to any of these.

Once you have established the conventions, you'd typically convert each rotation to a matrix then multiply them.